2016
DOI: 10.1007/s00605-016-0949-2
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Maximum volume polytopes inscribed in the unit sphere

Abstract: In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with d + 2 vertices in every dimension, and for polytopes with d + 3 vertices in odd dimensions. For polytopes with d + 3 vertices in even dimensions we give a partial solution.2010 Mathematics Subject Classification. 52B60, 52A40, 52A38.

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Cited by 14 publications
(22 citation statements)
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“…The first non-trivial case is when the number of points is equal to n = d + 2. It has been proved: 20]). Let P ∈ P d (d + 2) have maximal volume over P d (d + 2).…”
Section: Corollary 22 Implies the Following Onementioning
confidence: 93%
See 2 more Smart Citations
“…The first non-trivial case is when the number of points is equal to n = d + 2. It has been proved: 20]). Let P ∈ P d (d + 2) have maximal volume over P d (d + 2).…”
Section: Corollary 22 Implies the Following Onementioning
confidence: 93%
“…We now extract the method of Berman and Hanes to higher dimensions and using a combinatorial concept, the idea of Gale's transform solve some cases of few vertices. In this subsection we collect the results of the paper [20]. The first step is the generalization of Lemma 2.1 for arbitrary dimensions.…”
Section: Number Of Vertices Maximal Volume Number Of Facetsmentioning
confidence: 99%
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“…We solve a more general problem: we characterize the d-dimensional polytopes with n vertices whose geometric automorphism group contains D n as a subgroup. This result can be regarded as a first step towards solving Problem 1 [11], and we note that D n is the combinatorial automorphism group of a cyclic polytope in an even dimensional space.…”
Section: Z Lángimentioning
confidence: 91%
“…Nevertheless, according to our knowledge, this question, which is listed in both research problem books [2] and [3], is still open for polyhedra with n > 8 vertices apart from the fortunate case of n = 12 when the solution is the regular icosahedron. In [9] the authors investigated this problem for polytopes in arbitrary dimensions. By generalizing the methods of [1], presented a necessary condition for the optimality of a polytope.…”
Section: Introductionmentioning
confidence: 99%